Lesson 9 of 9 · 17 min
Chapter review
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Must-know facts
22 facts
- 01Stress = restoring force per unit area = F/A; unit N m⁻² = Pa; dimensions [ML⁻¹T⁻²].
- 02Strain is a ratio of lengths or volumes: no unit, no dimensions.
- 03Three strains: longitudinal ΔL/L, shearing Δx/L = tan θ ≈ θ, volume ΔV/V.
- 04Hooke's law: stress ∝ strain for small deformations; the constant is the modulus of elasticity.
- 05Stress-strain curve: O-A proportional (Hooke's law), B yield point or elastic limit (σy), C permanent set if unloaded, D ultimate tensile strength (σu), E fracture.
- 06D and E close: brittle. D and E far apart: ductile.
- 07Elastomers (rubber, aorta tissue): large elastic region, no Hooke's law over most of it, no well-defined plastic region.
- 08Y = FL/(AΔL); G = F/(Aθ); B = −p/(ΔV/V); all three have the unit Pa.
- 09Y and G apply only to solids; B applies to solids, liquids and gases.
- 10For most materials G ≈ Y/3.
- 11Y: steel 2.0 × 10¹¹, copper 1.1 × 10¹¹, bone 9.4 × 10⁹ N m⁻² (values used in NCERT's examples).
- 12Same 0.1% strain in a 0.1 cm² wire: steel 2000 N, copper 1100 N, brass 900 N, aluminium 690 N; steel is the most elastic.
- 13Compressibility k = 1/B; a gas squeezes roughly a million times more easily than a solid.
- 14Bulk modulus of water 2.2 × 10⁹ N m⁻²; air at STP 1.0 × 10⁵ N m⁻² (1.0 × 10⁻⁴ GPa).
- 15Water at 3000 m ocean depth (g = 10) is compressed by 1.36%.
- 16Poisson's ratio = lateral strain/longitudinal strain; unitless; steels 0.28-0.30, aluminium alloys about 0.33.
- 17Elastic energy U = ½ × stress × strain × volume; per unit volume u = ½σε = ½Y(strain)².
- 18Beam sag δ = Wl³/(4bd³Y): depth counts as d³, breadth only as b.
- 19Crane rope for 10 t: A ≥ Mg/σy = 3.3 × 10⁻⁴ m² (radius about 1 cm); with a safety factor of 10, radius about 3 cm, made of braided thin wires.
- 20Mountain height limit: hρg = 30 × 10⁷ N m⁻² with ρ = 3 × 10³ kg m⁻³ gives about 10 km.
- 21A wire hung from a ceiling with weight F at its end has tension F, not 2F, at every section, so the stress is F/A.
- 22Stress is not a vector: no single direction can be assigned to it.
Common traps
Where marks are lost
Formulas
12 to know
Stress
σ = F/A
Unit N m⁻² = Pa; [ML⁻¹T⁻²].
Strains
longitudinal ΔL/L; shearing Δx/L = tan θ ≈ θ; volume ΔV/V
All dimensionless.
Hooke's law
stress = k × strain
Valid only in the linear part OA.
Young's modulus
Y = σ/ε = (F/A)/(ΔL/L) = FL/(AΔL)
So ΔL = FL/(AY).
Shear modulus
G = (F/A)/(Δx/L) = FL/(AΔx) = F/(Aθ); σs = Gθ
G ≈ Y/3 for most materials.
Bulk modulus
B = −p/(ΔV/V)
Positive; applies to solids, liquids and gases.
Compressibility
k = 1/B = −(1/Δp)(ΔV/V)
Largest for gases.
Poisson's ratio
(Δd/d)/(ΔL/L) = (Δd/ΔL)(L/d)
Pure number; steels 0.28-0.30.
Elastic energy
U = ½ × stress × strain × volume = ½ Y A l²/L
Per unit volume u = ½σε.
Beam sag
δ = Wl³/(4bd³Y)
Beam supported near its ends, load W at the centre.
Minimum rope area
A ≥ Mg/σy
Keeps the rope within its elastic limit.
Mountain height limit
hρg = elastic limit of rock
30 × 10⁷ N m⁻², ρ = 3 × 10³ kg m⁻³, g = 10 m s⁻² give h = 10 km.
Key terms
15 terms
- Elasticity
- The tendency of a body to recover its original size and shape once the deforming force is removed.
- Plasticity
- The behaviour of a body that stays deformed after the force is removed, as putty and mud do.
- Stress
- Internal restoring force per unit area of a deformed body; equal in size to the applied force per unit area.
- Strain
- Change in a dimension divided by the original dimension; a pure number.
- Hydraulic stress
- Uniform normal stress from a surrounding fluid; equal to the fluid pressure and changing only volume.
- Elastic limit (yield point)
- Point B of the stress-strain curve, the largest stress after which the body still recovers fully.
- Permanent set
- The strain left in a body after unloading from beyond its yield point.
- Ultimate tensile strength
- The highest stress on the stress-strain curve, point D; past it the wire keeps stretching under a smaller force until it breaks.
- Brittle
- A material whose fracture point lies close to its ultimate strength.
- Ductile
- A material that strains a long way between its ultimate strength and fracture.
- Elastomer
- A substance such as rubber or aorta tissue that can take large strains and recover, without following Hooke's law.
- Modulus of rigidity
- Another name for the shear modulus G.
- Compressibility
- Reciprocal of bulk modulus; fractional volume change per unit rise in pressure.
- Poisson's ratio
- Lateral strain divided by longitudinal strain for a stretched wire.
- Buckling
- Sideways bending of a deep, thin bar under a load that is not exactly placed.